Near optimal holdem play with one round of betting

Consider the following simplified version of Texas holdem, with two players Alice and Bob: Alice and Bob are each dealt two private cards. Alice posts a small blind of 1, Bob posts a big blind of 2. Alice either folds, calls, or raises by any amount $\ge 2$. Bob either calls or folds. Five more shared cards are dealt, and the winner is determined as usual.

Exact poker win probabilities

Brief summary in case you just want the data: here’s a table of exact win/loss/tie probabilities for every pair of two card preflop hands in Texas holdem: exact.txt Eugene d’Eon and I have been playing around computing Nash equilibria for extremely simplified versions of heads-up Texas holdem poker. For those who don’t know the details, in Texas holdem each player is dealt two cards face down, followed by five cards face up which are shared between all players (with betting at various points in between these cards).

Minimal continuous poker

Consider the following poker-like game, played with two players: Alice and Bob. Bob posts a blind of 1. Both players are dealt a single, continuous hand chosen uniformly at random from $[0,1]$. Alice can fold, call, or raise any amount $b \gt 0$ (calling means $b = 0$). Bob either calls or folds. My original plan was to work out the Nash equilibrium for this game, and therefore derive interesting smooth curves describing the optimal way for Alice to bluff and generally obscure her hand.